A high-power Liquid-Lithium Target (LiLiT) was used for the first time for neutron production via the thick-target 7Li(p,n)7Be reaction and quantitative determination of neutron capture cross sections. Bombarded with a 1-2 mA proton beam at 1.92 MeV from the Soreq Applied Research Accelerator Facility (SARAF), the setup yields a 30-keV quasi-Maxwellian neutron spectrum with an intensity of 3-5e10 n/s, more than one order of magnitude larger than present near-threshold 7Li(p,n) neutron sources. The setup was used here to determine the 30-keV Maxwellian averaged cross section (MACS) of 94Zr and 96Zr as 28.0+-0.6 mb and 12.4+-0.5 mb respectively, based on activation measurements. The precision of the cross section determinations results both from the high neutron yield and from detailed simulations of the entire experimental setup. We plan to extend our experimental studies to low-abundance and radioactive targets. In addition, we show here that the setup yields intense high-energy (17.6 and 14.6 MeV) prompt capture gamma-rays from the 7Li(p,gamma)8Be reaction with yields of ~3e8 gammas/s/mA and ~4e8 gammas/s/mA, respectively, evidenced by the 90Zr(gamma,n)89Zr photonuclear reaction.